Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Suppose that we are given two independent sets I_b and I_r of a graph such that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I_b into I_r so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown, that is, it is in P, NP-complete, or PSPACE-complete in certain conditions. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours

Reconfiguring Triangulations

The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis: 1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem. 2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in. 3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e' assigns the same label to e' as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance

On Reconfiguration Problems: Structure and Tractability

Given an n-vertex graph G and two vertices s and t in G, determining whether there exists a path and computing the length of the shortest path between s and t are two of the most fundamental graph problems. In the classical battle of P versus NP or ``easy'' versus ``hard'', both of these problems are on the easy side. That is, they can be solved in poly(n) time, where poly is any polynomial function. But what if our input consisted of a 2^n-vertex graph? Of course, we can no longer assume G to be part of the input, as reading the input alone requires more than poly(n) time. Instead, we are given an oracle encoded using poly(n) bits and that can, in constant or poly(n) time, answer queries of the form ``is u a vertex in G'' or ``is there an edge between u and v?''. Given such an oracle and two vertices of the 2^n-vertex graph, can we still determine if there is a path or compute the length of the shortest path between s and t in poly(n) time? A slightly different, but equally insightful, formulation of the question above is as follows. Given a set S of n objects, consider the graph R(S) which contains one vertex for each set in the power set of S, 2^S, and two vertices are adjacent in R(S) whenever the size of their symmetric difference is equal to one. Clearly, this graph contains 2^n vertices and can be easily encoded in poly(n) bits using the oracle described above. It is not hard to see that there exists a path between any two vertices of R(S). Moreover, computing the length of a shortest path can be accomplished in constant time; it is equal to the size of the symmetric difference of the two underlying sets. If the vertex set of R(S) were instead restricted to a subset of 2^S, both of our problems can become NP-complete or even PSPACE-complete. Therefore, another interesting question is whether we can determine what types of ``restriction'' on the vertex set of R(S) induce such variations in the complexity of the two problems. These two seemingly artificial questions are in fact quite natural and appear in many practical and theoretical problems. In particular, these are exactly the types of questions asked under the reconfiguration framework, the main subject of this thesis. Under the reconfiguration framework, instead of finding a feasible solution to some instance I of a search problem Q, we are interested in structural and algorithmic questions related to the solution space of Q. Naturally, given some adjacency relation A defined over feasible solutions of Q, size of the symmetric difference being one such relation, the solution space can be represented using a graph R_Q(I). R_Q(I) contains one vertex for each feasible solution of Q on instance I and two vertices share an edge whenever their corresponding solutions are adjacent under A. An edge in R_Q(I) corresponds to a reconfiguration step, a walk in R_Q(I) is a sequence of such steps, a reconfiguration sequence, and R_Q(I) is a reconfiguration graph. Studying problems related to reconfiguration graphs has received considerable attention in recent literature, the most popular problem being to determine whether there exists a reconfiguration sequence between two given feasible solutions; for most NP-complete problems, this problem has been shown to be PSPACE-complete. The purpose of our work is to embark on a systematic investigation of the tractability and structural properties of such problems under both classical and parameterized complexity assumptions. Parameterized complexity is another framework which has become an essential tool for researchers in computational complexity during the last two decades or so and one of its main goals is to provide a better explanation of why some hard problems (in a classical sense) can be in fact much easier than others. Hence, we are interested in what separates the tractable instances from the intractable ones and the fixed-parameter tractable instances from the fixed-parameter intractable ones. It is clear from the generic definition of reconfiguration problems that several factors affect their complexity status. Our work aims at providing a finer classification of the complexity of reconfiguration problems with respect to some of these factors, including the definition of the adjacency relation A, structural properties of the input instance I, structural properties of the reconfiguration graph, and the length of a reconfiguration sequence. As most of these factors can be numerically quantified, we believe that the investigation of reconfiguration problems under both parameterized and classical complexity assumptions will help us further understand the boundaries between tractability and intractability. We consider reconfiguration problems related to Satisfiability, Coloring, Dominating Set, Vertex Cover, Independent Set, Feedback Vertex Set, and Odd Cycle Transversal, and provide lower bounds, polynomial-time algorithms, and fixed-parameter tractable algorithms. In doing so, we answer some of the questions left open in recent work and push the known boundaries between tractable and intractable even closer. As a byproduct of our initiating work on parameterized reconfiguration problems, we present a generic adaptation of parameterized complexity techniques which we believe can be used as a starting point for studying almost any such problem